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DOI: 10.4230/LIPIcs.CCC.2024.14

Complexity of Robust Orbit Problems for Torus Actions and the abc-Conjecture

Authors: Peter Bürgisser, Mahmut Levent Doğan, Visu Makam, Michael Walter, and Avi Wigderson

Published in: LIPIcs, Volume 300, 39th Computational Complexity Conference (CCC 2024)

Abstract

When a group acts on a set, it naturally partitions it into orbits, giving rise to orbit problems. These are natural algorithmic problems, as symmetries are central in numerous questions and structures in physics, mathematics, computer science, optimization, and more. Accordingly, it is of high interest to understand their computational complexity. Recently, Bürgisser et al. (2021) gave the first polynomial-time algorithms for orbit problems of torus actions, that is, actions of commutative continuous groups on Euclidean space. In this work, motivated by theoretical and practical applications, we study the computational complexity of robust generalizations of these orbit problems, which amount to approximating the distance of orbits in ℂⁿ up to a factor γ ≥ 1. In particular, this allows deciding whether two inputs are approximately in the same orbit or far from being so. On the one hand, we prove the NP-hardness of this problem for γ = n^Ω(1/log log n) by reducing the closest vector problem for lattices to it. On the other hand, we describe algorithms for solving this problem for an approximation factor γ = exp(poly(n)). Our algorithms combine tools from invariant theory and algorithmic lattice theory, and they also provide group elements witnessing the proximity of the given orbits (in contrast to the algebraic algorithms of prior work). We prove that they run in polynomial time if and only if a version of the famous number-theoretic abc-conjecture holds - establishing a new and surprising connection between computational complexity and number theory.

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Peter Bürgisser, Mahmut Levent Doğan, Visu Makam, Michael Walter, and Avi Wigderson. Complexity of Robust Orbit Problems for Torus Actions and the abc-Conjecture. In 39th Computational Complexity Conference (CCC 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 300, pp. 14:1-14:48, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{burgisser_et_al:LIPIcs.CCC.2024.14,
  author =	{B\"{u}rgisser, Peter and Do\u{g}an, Mahmut Levent and Makam, Visu and Walter, Michael and Wigderson, Avi},
  title =	{{Complexity of Robust Orbit Problems for Torus Actions and the abc-Conjecture}},
  booktitle =	{39th Computational Complexity Conference (CCC 2024)},
  pages =	{14:1--14:48},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-331-7},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{300},
  editor =	{Santhanam, Rahul},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2024.14},
  URN =		{urn:nbn:de:0030-drops-204100},
  doi =		{10.4230/LIPIcs.CCC.2024.14},
  annote =	{Keywords: computational invariant theory, geometric complexity theory, orbit problems, abc-conjecture, closest vector problem}
}

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Track A: Algorithms, Complexity and Games

DOI: 10.4230/LIPIcs.ICALP.2024.16

NP-Hardness of Testing Equivalence to Sparse Polynomials and to Constant-Support Polynomials

Authors: Omkar Baraskar, Agrim Dewan, Chandan Saha, and Pulkit Sinha

Published in: LIPIcs, Volume 297, 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024)

Abstract

An s-sparse polynomial has at most s monomials with nonzero coefficients. The Equivalence Testing problem for sparse polynomials (ETsparse) asks to decide if a given polynomial f is equivalent to (i.e., in the orbit of) some s-sparse polynomial. In other words, given f ∈ 𝔽[𝐱] and s ∈ ℕ, ETsparse asks to check if there exist A ∈ GL(|𝐱|, 𝔽) and 𝐛 ∈ 𝔽^|𝐱| such that f(A𝐱 + 𝐛) is s-sparse. We show that ETsparse is NP-hard over any field 𝔽, if f is given in the sparse representation, i.e., as a list of nonzero coefficients and exponent vectors. This answers a question posed by Gupta, Saha and Thankey (SODA 2023) and also, more explicitly, by Baraskar, Dewan and Saha (STACS 2024). The result implies that the Minimum Circuit Size Problem (MCSP) is NP-hard for a dense subclass of depth-3 arithmetic circuits if the input is given in sparse representation. We also show that approximating the smallest s₀ such that a given s-sparse polynomial f is in the orbit of some s₀-sparse polynomial to within a factor of s^{1/3 - ε} is NP-hard for any ε > 0; observe that s-factor approximation is trivial as the input is s-sparse. Finally, we show that for any constant σ ≥ 6, checking if a polynomial (given in sparse representation) is in the orbit of some support-σ polynomial is NP-hard. Support of a polynomial f is the maximum number of variables present in any monomial of f. These results are obtained via direct reductions from the 3-SAT problem.

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Omkar Baraskar, Agrim Dewan, Chandan Saha, and Pulkit Sinha. NP-Hardness of Testing Equivalence to Sparse Polynomials and to Constant-Support Polynomials. In 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 297, pp. 16:1-16:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{baraskar_et_al:LIPIcs.ICALP.2024.16,
  author =	{Baraskar, Omkar and Dewan, Agrim and Saha, Chandan and Sinha, Pulkit},
  title =	{{NP-Hardness of Testing Equivalence to Sparse Polynomials and to Constant-Support Polynomials}},
  booktitle =	{51st International Colloquium on Automata, Languages, and Programming (ICALP 2024)},
  pages =	{16:1--16:21},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-322-5},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{297},
  editor =	{Bringmann, Karl and Grohe, Martin and Puppis, Gabriele and Svensson, Ola},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2024.16},
  URN =		{urn:nbn:de:0030-drops-201598},
  doi =		{10.4230/LIPIcs.ICALP.2024.16},
  annote =	{Keywords: Equivalence testing, MCSP, sparse polynomials, 3SAT}
}

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DOI: 10.4230/LIPIcs.ECRTS.2017.2

Contention-Aware Dynamic Memory Bandwidth Isolation with Predictability in COTS Multicores: An Avionics Case Study

Authors: Ankit Agrawal, Gerhard Fohler, Johannes Freitag, Jan Nowotsch, Sascha Uhrig, and Michael Paulitsch

Published in: LIPIcs, Volume 76, 29th Euromicro Conference on Real-Time Systems (ECRTS 2017)

Abstract

Airbus is investigating COTS multicore platforms for safety-critical avionics applications, pursuing helicopter-style autonomous and electric aircraft. These aircraft need to be ultra-lightweight for future mobility in the urban city landscape. As a step towards certification, Airbus identified the need for new methods that preserve the ARINC 653 single core schedule of a Helicopter Terrain Awareness and Warning System (HTAWS) application while scheduling additional safety-critical partitions on the other cores. As some partitions in the HTAWS application are memory-intensive, static memory bandwidth throttling may lead to slow down of such partitions or provide only little remaining bandwidth to the other cores. Thus, there is a need for dynamic memory bandwidth isolation. This poses new challenges for scheduling, as execution times and scheduling become interdependent: scheduling requires execution times as input, which depends on memory latencies and contention from memory accesses of other cores - which are determined by scheduling. Furthermore, execution times depend on memory access patterns. In this paper, we propose a method to solve this problem for slot-based time-triggered systems without requiring application source-code modifications using a number of dynamic memory bandwidth levels. It is NoC and DRAM controller contention-aware and based on the existing interference-sensitive WCET computation and the memory bandwidth throttling mechanism. It constructs schedule tables by assigning partitions and dynamic memory bandwidth to each slot on each core, considering worst case memory access patterns. Then at runtime, two servers - for processing time and memory bandwidth - run on each core, jointly controlling the contention between the cores and the amount of memory accesses per slot. As a proof-of-concept, we use a constraint solver to construct tables. Experiments on the P4080 COTS multicore platform, using a research OS from Airbus and EEMBC benchmarks, demonstrate that our proposed method enables preserving existing schedules on a core while scheduling additional safety-critical partitions on other cores, and meets dynamic memory bandwidth isolation requirements.

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Ankit Agrawal, Gerhard Fohler, Johannes Freitag, Jan Nowotsch, Sascha Uhrig, and Michael Paulitsch. Contention-Aware Dynamic Memory Bandwidth Isolation with Predictability in COTS Multicores: An Avionics Case Study. In 29th Euromicro Conference on Real-Time Systems (ECRTS 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 76, pp. 2:1-2:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{agrawal_et_al:LIPIcs.ECRTS.2017.2,
  author =	{Agrawal, Ankit and Fohler, Gerhard and Freitag, Johannes and Nowotsch, Jan and Uhrig, Sascha and Paulitsch, Michael},
  title =	{{Contention-Aware Dynamic Memory Bandwidth Isolation with Predictability in COTS Multicores: An Avionics Case Study}},
  booktitle =	{29th Euromicro Conference on Real-Time Systems (ECRTS 2017)},
  pages =	{2:1--2:22},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-037-8},
  ISSN =	{1868-8969},
  year =	{2017},
  volume =	{76},
  editor =	{Bertogna, Marko},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ECRTS.2017.2},
  URN =		{urn:nbn:de:0030-drops-71740},
  doi =		{10.4230/LIPIcs.ECRTS.2017.2},
  annote =	{Keywords: Dynamic memory bandwidth isolation, Safety-critical avionics, COTS multicores}
}

3 Search Results for "Agrawal, Ankit"

Complexity of Robust Orbit Problems for Torus Actions and the abc-Conjecture

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NP-Hardness of Testing Equivalence to Sparse Polynomials and to Constant-Support Polynomials

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Contention-Aware Dynamic Memory Bandwidth Isolation with Predictability in COTS Multicores: An Avionics Case Study

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